# Browsing by Subject "math.DS"

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Item Open Access A Large Deviation Approach to Posterior Consistency in Dynamical SystemsSu, Langxuan; Mukherjee, SayanIn this paper, we provide asymptotic results concerning (generalized) Bayesian inference for certain dynamical systems based on a large deviation approach. Given a sequence of observations $y$, a class of model processes parameterized by $\theta \in \Theta$ which can be characterized as a stochastic process $X^\theta$ or a measure $\mu_\theta$, and a loss function $L$ which measures the error between $y$ and a realization of $X^\theta$, we specify the generalized posterior distribution $\pi_t(\theta \mid y)$. The goal of this paper is to study the asymptotic behavior of $\pi_t(\theta \mid y)$ as $t \to \infty.$ In particular, we state conditions on the model family $\{\mu_\theta\}_{\theta \in \Theta}$ and the loss function $L$ such that the posterior distribution converges. The two conditions we require are: (1) a conditional large deviation behavior for a single $X^\theta$, and (2) an exponential continuity condition over the model family for the map from the parameter $\theta$ to the loss incurred between $X^\theta$ and the observation sequence $y$. The proposed framework is quite general, we apply it to two very different classes of dynamical systems: continuous time hypermixing processes and Gibbs processes on shifts of finite type. We also show that the generalized posterior distribution concentrates asymptotically on those parameters that minimize the expected loss and a divergence term, hence proving posterior consistency.Item Open Access Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials(2017-11-30) Herzog, DP; Mattingly, JCWe study Langevin dynamics of $N$ particles on $R^d$ interacting through a singular repulsive potential, e.g.~the well-known Lennard-Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the main result relies on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions.Item Open Access Geometric ergodicity of Langevin dynamics with Coulomb interactionsLu, Y; Mattingly, JCThis paper is concerned with the long time behavior of Langevin dynamics of {\em Coulomb gases} in $\mathbf{R}^d$ with $d\geq 2$, that is a second order system of Brownian particles driven by an external force and a pairwise repulsive Coulomb force. We prove that the system converges exponentially to the unique Boltzmann-Gibbs invariant measure under a weighted total variation distance. The proof relies on a novel construction of Lyapunov function for the Coulomb system.Item Open Access Higher order asymptotics for large deviations -- Part IFernando, K; Hebbar, PFor sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth Expansions for the Central Limit Theorem. We apply our results to show that Diophantine iid sequences, finite state Markov chains, strongly ergodic Markov chains and Birkhoff sums of smooth expanding maps & subshifts of finite type satisfy these strong large deviation results.Item Open Access Noise-induced strong stabilizationLeimbach, Matti; Mattingly, Jonathan C; Scheutzow, MichaelWe consider a 2-dimensional stochastic differential equation in polar coordinates depending on several parameters. We show that if these parameters belong to a specific regime then the deterministic system explodes in finite time, but the random dynamical system corresponding to the stochastic equation is not only strongly complete but even admits a random attractor.Item Open Access On the mean-field limit for the Vlasov-Poisson-Fokker-Planck systemHuang, H; Liu, JG; Pickl, PWe devise and study a random particle blob method for approximating the Vlasov-Poisson-Fokkker-Planck (VPFP) equations by a $N$-particle system subject to the Brownian motion in $\mathbb{R}^3$ space. More precisely, we show that maximal distance between the exact microscopic and the mean-field trajectories is bounded by $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\leq\varepsilon<\frac{1}{36}$) for a system with blob size $N^{-\delta}$ ($\frac{1}{3}\leq\delta<\frac{19}{54}-\frac{2\varepsilon}{3}$) up to a probability $1-N^{-\alpha}$ for any $\alpha>0$, which improves the cut-off in [10]. Our result thus leads to a derivation of VPFP equations from the microscopic $N$-particle system. In particular we prove the convergence rate between the empirical measure associated to the particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates crucially rely on the randomness coming from the initial data and from the Brownian motion.Item Open Access Propagation of Fluctuations in Biochemical Systems, II: Nonlinear ChainsAnderson, DF; Mattingly, Jonathan ChristopherWe consider biochemical reaction chains and investigate how random external fluctuations, as characterized by variance and coefficient of variation, propagate down the chains. We perform such a study under the assumption that the number of molecules is high enough so that the behavior of the concentrations of the system is well approximated by differential equations. We conclude that the variances and coefficients of variation of the fluxes will decrease as one moves down the chain and, through an example, show that there is no corresponding result for the variances of the chemical species. We also prove that the fluctuations of the fluxes as characterized by their time averages decrease down reaction chains. The results presented give insight into how biochemical reaction systems are buffered against external perturbations solely by their underlying graphical structure and point out the benefits of studying the out-of-equilibrium dynamics of systems.Item Open Access Random Splitting of Fluid Models: Ergodicity and Convergence(2022-01-17) Agazzi, A; Mattingly, JC; Melikechi, OItem Open Access Random Splitting of Fluid Models: Positive Lyapunov Exponents(2022-10-06) Agazzi, Andrea; Mattingly, Jonathan C; Melikechi, OmarItem Open Access Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations(2017-07-27) Glatt-Holtz, NE; Herzog, DP; Mattingly, JCWe establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode control problem in a number of concrete nonlinear partial differential equations. We also develop applications concerning associated classes of stochastic partial differential equations (SPDEs). In particular, we study the support properties of probability laws corresponding to these SPDEs as well as provide applications concerning the ergodic and mixing properties of invariant measures for these stochastic systems.Item Open Access Scaling limits of a model for selection at two scales(2015) Luo, S; Mattingly, JCThe dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.Item Open Access Singularities of invariant densities for random switching between two linear ODEs in 2DBakhtin, Y; Hurth, T; Lawley, SD; Mattingly, JCWe consider a planar dynamical system generated by two stable linear vector fields with distinct fixed points and random switching between them. We characterize singularities of the invariant density in terms of the switching rates and contraction rates. We prove boundedness away from those singularities. We also discuss some motivating biological examples.Item Open Access Smooth invariant densities for random switching on the torus(2017-08-30) Bakhtin, Y; Hurth, T; Lawley, SD; Mattingly, JCWe consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal to each other at all points of the torus and that each of them allows for a smooth invariant density and no periodic orbits, we prove that the switched system also has a smooth invariant density, for every switching rate. Our approach is based on an integration by parts formula inspired by techniques from Malliavin calculus.